This is an equation from my paper Quantum Mechanics near Closed Time-like Lines which appeared in Physical Review D44, 3197–3217 (1991). It is the equation for the state of a quantum computer that is, by whatever means, provided with a method of sending its output back in time to interact with its input . The universality of quantum computation ensures that the results also apply to any time-travelling physical system, not just a computer. The self-consistency of this equation proves the self-consistency of time travel in quantum physics. Analysis of the equation shows that if we had a time machine and tried to use it to enact ‘paradoxes’ (like going back in time to prevent our building the time machine), we would simply go back to a universe in which those events really happened. U and U represent the motion – any motion we like – that we would initiate when we emerge from the time machine. ‘Tr’ represents the operation of selecting which part of the computer’s state it will send through the time machine. And represents the combining of the input information with that which has travelled back in time.


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